# American Institute of Mathematical Sciences

October  2012, 17(7): 2299-2311. doi: 10.3934/dcdsb.2012.17.2299

## A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies

 1 Departamento de Matemática, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Pabellón I, (1428) Buenos Aires 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong 3 Australian Research Council Centre & Excellence for Mathematics & Statistics of Complex Systems, School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia

Received  December 2011 Revised  April 2012 Published  July 2012

In prior work, a series of two-point boundary value problems have been investigated for a steady state two-ion electro-diffusion model system in which the sum of the valencies $\nu_+$ and $\nu_-$ is zero. In that case, reduction is obtained to the canonical Painlevé II equation for the scaled electric field. Here, a physically important Neumann boundary value problem in the generic case when $\nu_+ + \nu_-\neq 0$ is investigated. The problem is novel in that the model equation for the electric field involves yet to be determined boundary values of the solution. A reduction of the Neumann boundary value problem in terms of elliptic functions is obtained for privileged valency ratios. A topological index argument is used to establish the existence of a solution in the general case, under the assumption $\nu_+ + \nu_- \leq 0$.
Citation: Pablo Amster, Man Kam Kwong, Colin Rogers. A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2299-2311. doi: 10.3934/dcdsb.2012.17.2299
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